Find the values of x for which each function is continuous.
1.) f(x)= 2x+1/x^2+x-2
2.) f(x)= x if x is less than or =1
= 2x-1 if x>1
To find the values of x for which each function is continuous, we need to consider the conditions required for continuity.
1.) f(x) = (2x + 1)/(x^2 + x - 2)
To determine the values of x for which this function is continuous, we need to examine two aspects:
a) The denominator of the fraction: x^2 + x - 2. A rational function is continuous for all values of x except where the denominator is equal to zero. So, we need to find the values of x that make the denominator zero.
Setting x^2 + x - 2 = 0, we can factor this quadratic equation as (x - 1)(x + 2) = 0. Solving for x, we get x = 1 and x = -2. These are the values we need to exclude from the domain of the function for continuity.
b) The numerator of the fraction: 2x + 1. As a linear function, there are no restrictions on x for it to be continuous.
Therefore, the values of x for which the function f(x) = (2x + 1)/(x^2 + x - 2) is continuous are all real numbers except x = 1 and x = -2.
2.) f(x) = x, if x ≤ 1
= 2x - 1, if x > 1
For this piecewise function, we need to examine the continuity at x = 1 separately since it is the point where the two cases meet.
a) x ≤ 1:
In this case, f(x) = x. A linear function is continuous for all real numbers.
b) x > 1:
In this case, f(x) = 2x - 1. Again, a linear function is continuous for all real numbers.
c) At the point where the two cases meet, x = 1:
To check continuity at x = 1, we need to ensure that the limit of f(x) as x approaches 1 from both sides is equal. Taking the limit as x approaches 1^- (from the left) and as x approaches 1^+ (from the right), we find that both limits equal 1.
Therefore, the function f(x) = x, if x ≤ 1, and f(x) = 2x - 1, if x > 1, is continuous for all real numbers.